I'm an experimental physicisist and not a theorist, so I may not the most qualified person to answer
your question. So although I'll do my best, please take my answer with a grain of salt.

In researching your question I came across this paper with a pretty concise formulation of the decoupling theorem. For a
theory containing two scalar fields (each with an associated Higgs-like boson), and with the mass
scales (Higgs boson masses) well-separated (i.e. one is much more massive than the other), the
theorem states:

"At scales below the mass of the heavy particle the full theory may be approximated arbitrarily
closely by an effective theory of the light particle alone, with naturalness scale the heavy particle
mass."

So what does this mean, and why does it matter? Well, the Higgs Boson that we discovered recently at
CERN poses a challenge for theorists, because (so far) it appears to behave more or less exactly like
the single Higgs predicted in the Standard Model (or SM, for short) of
particle physics. And with the observed Higgs mass of about 126 GeV, the SM can work as-is up to
much higher energy ranges, without the need for new physics.

And that's a problem, because the SM is known to be incomplete. Gravitation does not fit in the SM,
nor is there any explanation for dark matter, neutrinos with non-zero masses, etc. Theorists have
been working for many years to come up with a more complete model, and there are several
"Beyond the Standard Model" theories, with Supersymmetry (SUSY) one of the most-
studied. All of these beyond-the-standard-model theories predict the existence of new particles and
processes at higher energies. And they contain alternatives or extensions to the single-Higgs
mechanism of the SM.

So with no new particles or phenomena appearing (so far) in LHC data, and the observed Higgs mass
and properties in agreement (again, so far) with the Standard Model, what is a theorist to do? Well,
the decoupling theorem allows some wiggle room. Different versions of SUSY (for example) include
one or more Higgs pairs, or 'doublets'. A single Higgs doesn't fit in SUSY, but what if our 126 GeV
Higgs is just half of a doublet, with the second boson much heavier than the one we found? The
decoupling theorem says that at lower energies the physics will look just like a single-Higgs theory.
But at higher energy scales, both Higgs doublet members become important, as well as other new
physics.

And what does it mean for us experimentalists? Well, now that we have found our Higgs boson, the
next
step at LHC (and the future Next Linear Collider) is to study its properties as carefully as possible.
Any observed deviations from the Standard Model predictions may give us clues about the existence
of additional Higgs bosons and/or other new physics at higher energy scales than we can directly
observe.

I hope this was useful for you, and that I'm not too far off the mark. I'll end this with a link to a
seminar by
Howard Haber. A good deal of his talk is pretty high level, but on the second-but-last side, where he
wraps up his discussion of decoupling, his final conclusions pretty much summarize what I have
tried to say above:

The decoupling limit is a generic feature of extended Higgs sectors.

The observation of a SM-like Higgs boson does not rule out the possibility of an extended Higgs
sector in the decoupling regime.

Deviations from SM-Higgs behavior would provide clues to the structure of the extended Higgs
sector and/or the structure of physics beyond the SM.